Problem: Simplify the following expression: $ r = \dfrac{5}{8} + \dfrac{-10k - 9}{3k + 5} $
In order to add expressions, they must have a common denominator. Multiply the first expression by $\dfrac{3k + 5}{3k + 5}$ $ \dfrac{5}{8} \times \dfrac{3k + 5}{3k + 5} = \dfrac{15k + 25}{24k + 40} $ Multiply the second expression by $\dfrac{8}{8}$ $ \dfrac{-10k - 9}{3k + 5} \times \dfrac{8}{8} = \dfrac{-80k - 72}{24k + 40} $ Therefore $ r = \dfrac{15k + 25}{24k + 40} + \dfrac{-80k - 72}{24k + 40} $ Now the expressions have the same denominator we can simply add the numerators: $r = \dfrac{15k + 25 - 80k - 72}{24k + 40} $ $r = \dfrac{-65k - 47}{24k + 40}$